The Prokofiev-Svistunov-Ising process is rapidly mixing Tim Garoni - - PowerPoint PPT Presentation

Introduction Main Theorem Proof Discussion

The Prokofiev-Svistunov-Ising process is rapidly mixing

Tim Garoni

School of Mathematical Sciences Monash University

Introduction Main Theorem Proof Discussion

Collaborators

◮ Andrea Collevecchio (Monash University) ◮ Tim Hyndman (Monash University) ◮ Daniel Tokarev (Monash University)

Introduction Main Theorem Proof Discussion

Ising model - Motivation

◮ Introduced in 1920 as a model for ferromagnetism ◮ Hope was to explain the Curie transition:

◮ Place iron in a magnetic field ◮ Increase field to high value ◮ Slowly reduce field to zero ◮ Exists critical temperature Tc below which iron remains magnetized

Introduction Main Theorem Proof Discussion

Ising model - Motivation

◮ Introduced in 1920 as a model for ferromagnetism ◮ Hope was to explain the Curie transition:

◮ Place iron in a magnetic field ◮ Increase field to high value ◮ Slowly reduce field to zero ◮ Exists critical temperature Tc below which iron remains magnetized

◮ During mid 1930s it was realized the Ising model also describes

phases transitions in other physical systems

◮ Gas-liquid critical phenomena (lattice gas) ◮ Binary alloys

Introduction Main Theorem Proof Discussion

Ising model - Motivation

◮ Introduced in 1920 as a model for ferromagnetism ◮ Hope was to explain the Curie transition:

◮ Place iron in a magnetic field ◮ Increase field to high value ◮ Slowly reduce field to zero ◮ Exists critical temperature Tc below which iron remains magnetized

◮ During mid 1930s it was realized the Ising model also describes

phases transitions in other physical systems

◮ Gas-liquid critical phenomena (lattice gas) ◮ Binary alloys

◮ Now a paradigm for order/disorder transitions with applications to

◮ Economics (opinion formation) ◮ Finance (stock price dynamics) ◮ Biology (hemoglobin, DNA, . . . ) ◮ Image processing (archetypal Markov random field)

Introduction Main Theorem Proof Discussion

Ising model - Motivation

◮ Introduced in 1920 as a model for ferromagnetism ◮ Hope was to explain the Curie transition:

◮ Place iron in a magnetic field ◮ Increase field to high value ◮ Slowly reduce field to zero ◮ Exists critical temperature Tc below which iron remains magnetized

◮ During mid 1930s it was realized the Ising model also describes

phases transitions in other physical systems

◮ Gas-liquid critical phenomena (lattice gas) ◮ Binary alloys

◮ Now a paradigm for order/disorder transitions with applications to

◮ Economics (opinion formation) ◮ Finance (stock price dynamics) ◮ Biology (hemoglobin, DNA, . . . ) ◮ Image processing (archetypal Markov random field)

◮ Continues to play fundamental role in theoretical/mathematical

studies of phase transitions and critical phenomena

Introduction Main Theorem Proof Discussion

The Ising model

◮ Finite graph G = (V, E) ◮ Configuration space ΣG = {−1, +1}V ◮ Measure

P(σ) = 1 Z exp  β

• ij∈E

σiσj + h

• i∈V

σi  

◮ Inverse temperature β ◮ External field h ◮ Z is the partition function

Introduction Main Theorem Proof Discussion

The Ising model

◮ Finite graph G = (V, E) ◮ Configuration space ΣG = {−1, +1}V ◮ Measure

P(σ) = 1 Z exp  β

• ij∈E

σiσj + h

• i∈V

σi  

◮ Inverse temperature β ◮ External field h ◮ Z is the partition function ◮ Main physical interest is in certain expectations such as:

◮ Two-point correlation cov(σu, σv) = E(σuσv) − E(σu)E(σv) ◮ Susceptibility χ =

1 |V |

• u,v∈V

cov(σu, σv)

Introduction Main Theorem Proof Discussion

Phase transition

Let Λn = {−n, . . . , n}d ⊂ Zd, and consider Σ+

Λn = {σ ∈ {−1, 1}Zd : σi = +1 for all i ∈ Λn}

Sequence of Gibbs measures on Λn converges: P+

Λn,β,h ⇒ P+ β,h

Introduction Main Theorem Proof Discussion

Phase transition

Let Λn = {−n, . . . , n}d ⊂ Zd, and consider Σ+

Λn = {σ ∈ {−1, 1}Zd : σi = +1 for all i ∈ Λn}

Sequence of Gibbs measures on Λn converges: P+

Λn,β,h ⇒ P+ β,h

Analogous construction for “minus” boundary conditions

Introduction Main Theorem Proof Discussion

Phase transition

Let Λn = {−n, . . . , n}d ⊂ Zd, and consider Σ+

Λn = {σ ∈ {−1, 1}Zd : σi = +1 for all i ∈ Λn}

Sequence of Gibbs measures on Λn converges: P+

Λn,β,h ⇒ P+ β,h

Analogous construction for “minus” boundary conditions

Theorem (Aizenman, Duminil-Copin, Sidoravicius (2014))

• 1. If d = 1, then for any (β, h) ∈ [0, ∞) × R there is a unique

infinite-volume Gibbs measure

Introduction Main Theorem Proof Discussion

Phase transition

Let Λn = {−n, . . . , n}d ⊂ Zd, and consider Σ+

Λn = {σ ∈ {−1, 1}Zd : σi = +1 for all i ∈ Λn}

Sequence of Gibbs measures on Λn converges: P+

Λn,β,h ⇒ P+ β,h

Analogous construction for “minus” boundary conditions

Theorem (Aizenman, Duminil-Copin, Sidoravicius (2014))

• 1. If d = 1, then for any (β, h) ∈ [0, ∞) × R there is a unique

infinite-volume Gibbs measure

• 2. If d ≥ 2 and h = 0, then for any β ∈ [0, ∞) there is a unique

infinite-volume Gibbs measure

Introduction Main Theorem Proof Discussion

Phase transition

Let Λn = {−n, . . . , n}d ⊂ Zd, and consider Σ+

Λn = {σ ∈ {−1, 1}Zd : σi = +1 for all i ∈ Λn}

Sequence of Gibbs measures on Λn converges: P+

Λn,β,h ⇒ P+ β,h

Analogous construction for “minus” boundary conditions

Theorem (Aizenman, Duminil-Copin, Sidoravicius (2014))

• 1. If d = 1, then for any (β, h) ∈ [0, ∞) × R there is a unique

infinite-volume Gibbs measure

• 2. If d ≥ 2 and h = 0, then for any β ∈ [0, ∞) there is a unique

infinite-volume Gibbs measure

• 3. If d ≥ 2 and h = 0, there exists βc(d) ∈ (0, ∞) such that:
• a. If β ≤ βc, there is a unique infinite-volume Gibbs measure
• b. If β > βc, then P+

β,0 = P− β,0

Introduction Main Theorem Proof Discussion

Exact solutions

◮ Gn = Zn solved by Ising (1925)

Introduction Main Theorem Proof Discussion

Exact solutions

◮ Gn = Zn solved by Ising (1925) ◮ Gn = Kn solved by Husimi (1953) and Temperley (1954)

Introduction Main Theorem Proof Discussion

Exact solutions

◮ Gn = Zn solved by Ising (1925) ◮ Gn = Kn solved by Husimi (1953) and Temperley (1954) ◮ Gn = Z2 n solved by Peierls (1936), Kramers & Wannier (1941),

Onsager (1944), Yang (1952), Kasteleyn (1963), Fisher (1966)

Introduction Main Theorem Proof Discussion

Exact solutions

◮ Gn = Zn solved by Ising (1925) ◮ Gn = Kn solved by Husimi (1953) and Temperley (1954) ◮ Gn = Z2 n solved by Peierls (1936), Kramers & Wannier (1941),

Onsager (1944), Yang (1952), Kasteleyn (1963), Fisher (1966)

Introduction Main Theorem Proof Discussion

Exact solutions

◮ Gn = Zn solved by Ising (1925) ◮ Gn = Kn solved by Husimi (1953) and Temperley (1954) ◮ Gn = Z2 n solved by Peierls (1936), Kramers & Wannier (1941),

Onsager (1944), Yang (1952), Kasteleyn (1963), Fisher (1966) Smirnov (2006) proved critical interfaces between +/− components have conformally invariant limit

◮ SLE(3) - Schramm-L¨

• wner Evolution

Introduction Main Theorem Proof Discussion

Exact solutions

◮ Gn = Zn solved by Ising (1925) ◮ Gn = Kn solved by Husimi (1953) and Temperley (1954) ◮ Gn = Z2 n solved by Peierls (1936), Kramers & Wannier (1941),

Onsager (1944), Yang (1952), Kasteleyn (1963), Fisher (1966) Smirnov (2006) proved critical interfaces between +/− components have conformally invariant limit

◮ SLE(3) - Schramm-L¨

• wner Evolution

K-F related Ising partition function to perfect matchings

Introduction Main Theorem Proof Discussion

Exact solutions

◮ Gn = Zn solved by Ising (1925) ◮ Gn = Kn solved by Husimi (1953) and Temperley (1954) ◮ Gn = Z2 n solved by Peierls (1936), Kramers & Wannier (1941),

Onsager (1944), Yang (1952), Kasteleyn (1963), Fisher (1966) Smirnov (2006) proved critical interfaces between +/− components have conformally invariant limit

◮ SLE(3) - Schramm-L¨

• wner Evolution

K-F related Ising partition function to perfect matchings

◮ Elegant solution on planar graphs in terms of Pfaffians

Introduction Main Theorem Proof Discussion

Exact solutions

◮ Gn = Zn solved by Ising (1925) ◮ Gn = Kn solved by Husimi (1953) and Temperley (1954) ◮ Gn = Z2 n solved by Peierls (1936), Kramers & Wannier (1941),

Onsager (1944), Yang (1952), Kasteleyn (1963), Fisher (1966) Smirnov (2006) proved critical interfaces between +/− components have conformally invariant limit

◮ SLE(3) - Schramm-L¨

• wner Evolution

K-F related Ising partition function to perfect matchings

◮ Elegant solution on planar graphs in terms of Pfaffians ◮ Only tractable on graphs of bounded (small) genus

Introduction Main Theorem Proof Discussion

Exact solutions

◮ Gn = Zn solved by Ising (1925) ◮ Gn = Kn solved by Husimi (1953) and Temperley (1954) ◮ Gn = Z2 n solved by Peierls (1936), Kramers & Wannier (1941),

Onsager (1944), Yang (1952), Kasteleyn (1963), Fisher (1966) Smirnov (2006) proved critical interfaces between +/− components have conformally invariant limit

◮ SLE(3) - Schramm-L¨

• wner Evolution

K-F related Ising partition function to perfect matchings

◮ Elegant solution on planar graphs in terms of Pfaffians ◮ Only tractable on graphs of bounded (small) genus ◮ Method not tractable for Gn = Zd

n with d ≥ 3

Introduction Main Theorem Proof Discussion

Computational Complexity

◮ PARTITION:

◮ Input: Finite graph G = (V, E), and parameters β, h ◮ Output: Ising partition function

◮ CORRELATION:

◮ Input: Finite graph G = (V, E), a pair u, v ∈ V , and parameters β, h ◮ Output: Ising two-point correlation function

◮ SUSCEPTIBILITY:

◮ Input: Finite graph G = (V, E), and parameters β, h ◮ Output: Ising susceptibility

Introduction Main Theorem Proof Discussion

Computational Complexity

◮ PARTITION:

◮ Input: Finite graph G = (V, E), and parameters β, h ◮ Output: Ising partition function

◮ CORRELATION:

◮ Input: Finite graph G = (V, E), a pair u, v ∈ V , and parameters β, h ◮ Output: Ising two-point correlation function

◮ SUSCEPTIBILITY:

◮ Input: Finite graph G = (V, E), and parameters β, h ◮ Output: Ising susceptibility

Proposition (Jerrum-Sinclair 1993; Sinclair-Srivastava 2014 )

PARTITION, SUSCEPTIBILITY and CORRELATION are #P-hard.

Introduction Main Theorem Proof Discussion

Markov-chain Monte Carlo

◮ Construct a transition matrix P on Ω which:

◮ Is ergodic ◮ Has stationary distribution π(·)

◮ Generate random samples with (approximate) distribution π ◮ Estimate π expectations using sample means

Introduction Main Theorem Proof Discussion

Markov-chain Monte Carlo

◮ Construct a transition matrix P on Ω which:

◮ Is ergodic ◮ Has stationary distribution π(·)

◮ Generate random samples with (approximate) distribution π ◮ Estimate π expectations using sample means

d(t) := max

x∈Ω P t(x, ·) − π(·) ≤ Cαt,

for α ∈ (0, 1)

◮ Mixing time quantifies the rate of convergence

tmix(δ) := min {t : d(t) ≤ δ}

Introduction Main Theorem Proof Discussion

Markov-chain Monte Carlo

◮ Construct a transition matrix P on Ω which:

◮ Is ergodic ◮ Has stationary distribution π(·)

◮ Generate random samples with (approximate) distribution π ◮ Estimate π expectations using sample means

d(t) := max

x∈Ω P t(x, ·) − π(·) ≤ Cαt,

for α ∈ (0, 1)

◮ Mixing time quantifies the rate of convergence

tmix(δ) := min {t : d(t) ≤ δ}

◮ How does tmix depend on size of Ω?

Introduction Main Theorem Proof Discussion

Markov-chain Monte Carlo

◮ Construct a transition matrix P on Ω which:

◮ Is ergodic ◮ Has stationary distribution π(·)

◮ Generate random samples with (approximate) distribution π ◮ Estimate π expectations using sample means

d(t) := max

x∈Ω P t(x, ·) − π(·) ≤ Cαt,

for α ∈ (0, 1)

◮ Mixing time quantifies the rate of convergence

tmix(δ) := min {t : d(t) ≤ δ}

◮ How does tmix depend on size of Ω?

◮ For Ising the size of a problem instance is n = |V | ◮ If tmix = O(poly(n)) we have rapid mixing

Introduction Main Theorem Proof Discussion

Markov-chain Monte Carlo

◮ Construct a transition matrix P on Ω which:

◮ Is ergodic ◮ Has stationary distribution π(·)

◮ Generate random samples with (approximate) distribution π ◮ Estimate π expectations using sample means

d(t) := max

x∈Ω P t(x, ·) − π(·) ≤ Cαt,

for α ∈ (0, 1)

◮ Mixing time quantifies the rate of convergence

tmix(δ) := min {t : d(t) ≤ δ}

◮ How does tmix depend on size of Ω?

◮ For Ising the size of a problem instance is n = |V | ◮ If tmix = O(poly(n)) we have rapid mixing ◮ |Ω| = 2n so rapid mixing implies only logarithmically-many states

need be visited to reach approximate stationarity

Introduction Main Theorem Proof Discussion

Markov chains for the Ising model

◮ Glauber process (arbitrary field) 1963

◮ Lubetzky & Sly (2012): Rapidly mixing on boxes in Z2 at h = 0 iff

β ≤ βc

◮ Levin, Luczak & Peres (2010): Precise asymptotics on Kn for h = 0 ◮ Not used by computational physicists

Introduction Main Theorem Proof Discussion

Markov chains for the Ising model

◮ Glauber process (arbitrary field) 1963

◮ Lubetzky & Sly (2012): Rapidly mixing on boxes in Z2 at h = 0 iff

β ≤ βc

◮ Levin, Luczak & Peres (2010): Precise asymptotics on Kn for h = 0 ◮ Not used by computational physicists

◮ Swendsen-Wang process (zero field) 1987

◮ Simulates coupling of Ising and Fortuin-Kasteleyn models ◮ Long, Nachmias, Ding & Peres (2014): Precise asymptotics on Kn ◮ Ullrich (2014): Rapidly mixing on boxes in Z2 at all β = βc ◮ Empirically fast. State of the art 1987 – recently

Introduction Main Theorem Proof Discussion

Markov chains for the Ising model

◮ Glauber process (arbitrary field) 1963

◮ Lubetzky & Sly (2012): Rapidly mixing on boxes in Z2 at h = 0 iff

β ≤ βc

◮ Levin, Luczak & Peres (2010): Precise asymptotics on Kn for h = 0 ◮ Not used by computational physicists

◮ Swendsen-Wang process (zero field) 1987

◮ Simulates coupling of Ising and Fortuin-Kasteleyn models ◮ Long, Nachmias, Ding & Peres (2014): Precise asymptotics on Kn ◮ Ullrich (2014): Rapidly mixing on boxes in Z2 at all β = βc ◮ Empirically fast. State of the art 1987 – recently

◮ Jerrum-Sinclair process (positive field) 1993

◮ Simulates high-temperature graphs for h > 0 ◮ Proved rapidly mixing on all graphs at all temperatures for all h > 0 ◮ No empirical results - not used by computational physicists

Introduction Main Theorem Proof Discussion

Markov chains for the Ising model

◮ Glauber process (arbitrary field) 1963

◮ Lubetzky & Sly (2012): Rapidly mixing on boxes in Z2 at h = 0 iff

β ≤ βc

◮ Levin, Luczak & Peres (2010): Precise asymptotics on Kn for h = 0 ◮ Not used by computational physicists

◮ Swendsen-Wang process (zero field) 1987

◮ Simulates coupling of Ising and Fortuin-Kasteleyn models ◮ Long, Nachmias, Ding & Peres (2014): Precise asymptotics on Kn ◮ Ullrich (2014): Rapidly mixing on boxes in Z2 at all β = βc ◮ Empirically fast. State of the art 1987 – recently

◮ Jerrum-Sinclair process (positive field) 1993

◮ Simulates high-temperature graphs for h > 0 ◮ Proved rapidly mixing on all graphs at all temperatures for all h > 0 ◮ No empirical results - not used by computational physicists

◮ Prokofiev-Svistunov worm process (zero field) 2001

◮ No rigorous results currently known ◮ Empirically, best method known for susceptibility (Deng, G., Sokal) ◮ Widely used by computational physicists

Introduction Main Theorem Proof Discussion

Mixing time bound for PS process

Theorem (Collevecchio, G., Hyndman, Tokarev 2014+)

For any temperature, the mixing time of the PS process on graph G = (V, E) satisfies tmix(δ) = O(∆(G)m2n5) with n = |V |, m = |E| and ∆(G) the maximum degree.

Introduction Main Theorem Proof Discussion

Mixing time bound for PS process

Theorem (Collevecchio, G., Hyndman, Tokarev 2014+)

For any temperature, the mixing time of the PS process on graph G = (V, E) satisfies tmix(δ) = O(∆(G)m2n5) with n = |V |, m = |E| and ∆(G) the maximum degree. Only Markov chain for the Ising model currently known to be rapidly mixing at the critical point for boxes in Zd

Introduction Main Theorem Proof Discussion

High-temperature expansions and the PS measure

◮ Let Ck = {A ⊆ E : (V, A) has k odd vertices}

Introduction Main Theorem Proof Discussion

High-temperature expansions and the PS measure

◮ Let Ck = {A ⊆ E : (V, A) has k odd vertices} ◮ Let CW = {A ⊆ E : W is the set of odd vertices in (V, A) }

Introduction Main Theorem Proof Discussion

High-temperature expansions and the PS measure

◮ Let Ck = {A ⊆ E : (V, A) has k odd vertices} ◮ Let CW = {A ⊆ E : W is the set of odd vertices in (V, A) } ◮ PS measure defined on the configuration space C0 ∪ C2

π(A) ∝ x|A|

• n,

A ∈ C0, 2, A ∈ C2.

Introduction Main Theorem Proof Discussion

High-temperature expansions and the PS measure

◮ Let Ck = {A ⊆ E : (V, A) has k odd vertices} ◮ Let CW = {A ⊆ E : W is the set of odd vertices in (V, A) } ◮ PS measure defined on the configuration space C0 ∪ C2

π(A) ∝ x|A|

• n,

A ∈ C0, 2, A ∈ C2.

◮ If x = tanh β then:

◮ Ising susceptibility χ =

1 π(C0)

◮ Ising two-point correlation function E(σuσv) = n

2 π(Cuv) π(C0)

Introduction Main Theorem Proof Discussion

High-temperature expansions and the PS measure

◮ Let Ck = {A ⊆ E : (V, A) has k odd vertices} ◮ Let CW = {A ⊆ E : W is the set of odd vertices in (V, A) } ◮ PS measure defined on the configuration space C0 ∪ C2

π(A) ∝ x|A|

• n,

A ∈ C0, 2, A ∈ C2.

◮ If x = tanh β then:

◮ Ising susceptibility χ =

1 π(C0)

◮ Ising two-point correlation function E(σuσv) = n

2 π(Cuv) π(C0)

◮ PS measure is stationary distribution of PS process

Introduction Main Theorem Proof Discussion

Fully-polynomial randomized approximation schemes

Definition

An fpras for an Ising property f is a randomized algorithm such that for given G, T, and ξ, η ∈ (0, 1) the output Y satisfies P[(1 − ξ)f ≤ Y ≤ (1 + ξ)f] ≥ 1 − η and the running time is bounded by a polynomial in n, ξ−1, η−1.

Introduction Main Theorem Proof Discussion

Fully-polynomial randomized approximation schemes

Definition

An fpras for an Ising property f is a randomized algorithm such that for given G, T, and ξ, η ∈ (0, 1) the output Y satisfies P[(1 − ξ)f ≤ Y ≤ (1 + ξ)f] ≥ 1 − η and the running time is bounded by a polynomial in n, ξ−1, η−1. Combine rapid mixing of PS process with general fpras construction

• f Jerrum-Sinclair (1993):

Introduction Main Theorem Proof Discussion

Fully-polynomial randomized approximation schemes

Definition

An fpras for an Ising property f is a randomized algorithm such that for given G, T, and ξ, η ∈ (0, 1) the output Y satisfies P[(1 − ξ)f ≤ Y ≤ (1 + ξ)f] ≥ 1 − η and the running time is bounded by a polynomial in n, ξ−1, η−1. Combine rapid mixing of PS process with general fpras construction

• f Jerrum-Sinclair (1993):

◮ Let A ⊆ C0 ∪ C2 with π(A) ≥ 1/S(n) for some polynomial S(n)

Introduction Main Theorem Proof Discussion

Fully-polynomial randomized approximation schemes

Definition

An fpras for an Ising property f is a randomized algorithm such that for given G, T, and ξ, η ∈ (0, 1) the output Y satisfies P[(1 − ξ)f ≤ Y ≤ (1 + ξ)f] ≥ 1 − η and the running time is bounded by a polynomial in n, ξ−1, η−1. Combine rapid mixing of PS process with general fpras construction

• f Jerrum-Sinclair (1993):

◮ Let A ⊆ C0 ∪ C2 with π(A) ≥ 1/S(n) for some polynomial S(n) ◮ The following defines an fpras for π(A):

Introduction Main Theorem Proof Discussion

Fully-polynomial randomized approximation schemes

Definition

An fpras for an Ising property f is a randomized algorithm such that for given G, T, and ξ, η ∈ (0, 1) the output Y satisfies P[(1 − ξ)f ≤ Y ≤ (1 + ξ)f] ≥ 1 − η and the running time is bounded by a polynomial in n, ξ−1, η−1. Combine rapid mixing of PS process with general fpras construction

• f Jerrum-Sinclair (1993):

◮ Let A ⊆ C0 ∪ C2 with π(A) ≥ 1/S(n) for some polynomial S(n) ◮ The following defines an fpras for π(A):

◮ Let R(G) be our upper bound for tmix(δ) with δ = ξ/[8S(n)]

Introduction Main Theorem Proof Discussion

Fully-polynomial randomized approximation schemes

Definition

An fpras for an Ising property f is a randomized algorithm such that for given G, T, and ξ, η ∈ (0, 1) the output Y satisfies P[(1 − ξ)f ≤ Y ≤ (1 + ξ)f] ≥ 1 − η and the running time is bounded by a polynomial in n, ξ−1, η−1. Combine rapid mixing of PS process with general fpras construction

• f Jerrum-Sinclair (1993):

◮ Let A ⊆ C0 ∪ C2 with π(A) ≥ 1/S(n) for some polynomial S(n) ◮ The following defines an fpras for π(A):

◮ Let R(G) be our upper bound for tmix(δ) with δ = ξ/[8S(n)] ◮ Let Y = 1A

Introduction Main Theorem Proof Discussion

Fully-polynomial randomized approximation schemes

Definition

An fpras for an Ising property f is a randomized algorithm such that for given G, T, and ξ, η ∈ (0, 1) the output Y satisfies P[(1 − ξ)f ≤ Y ≤ (1 + ξ)f] ≥ 1 − η and the running time is bounded by a polynomial in n, ξ−1, η−1. Combine rapid mixing of PS process with general fpras construction

• f Jerrum-Sinclair (1993):

◮ Let A ⊆ C0 ∪ C2 with π(A) ≥ 1/S(n) for some polynomial S(n) ◮ The following defines an fpras for π(A):

◮ Let R(G) be our upper bound for tmix(δ) with δ = ξ/[8S(n)] ◮ Let Y = 1A ◮ Run the PS process T = R(G) time steps and record YT

Introduction Main Theorem Proof Discussion

Fully-polynomial randomized approximation schemes

Definition

An fpras for an Ising property f is a randomized algorithm such that for given G, T, and ξ, η ∈ (0, 1) the output Y satisfies P[(1 − ξ)f ≤ Y ≤ (1 + ξ)f] ≥ 1 − η and the running time is bounded by a polynomial in n, ξ−1, η−1. Combine rapid mixing of PS process with general fpras construction

• f Jerrum-Sinclair (1993):

◮ Let A ⊆ C0 ∪ C2 with π(A) ≥ 1/S(n) for some polynomial S(n) ◮ The following defines an fpras for π(A):

◮ Let R(G) be our upper bound for tmix(δ) with δ = ξ/[8S(n)] ◮ Let Y = 1A ◮ Run the PS process T = R(G) time steps and record YT ◮ Independently generate 72ξ−2S(n) such samples and take the

sample mean

Introduction Main Theorem Proof Discussion

Fully-polynomial randomized approximation schemes

Definition

An fpras for an Ising property f is a randomized algorithm such that for given G, T, and ξ, η ∈ (0, 1) the output Y satisfies P[(1 − ξ)f ≤ Y ≤ (1 + ξ)f] ≥ 1 − η and the running time is bounded by a polynomial in n, ξ−1, η−1. Combine rapid mixing of PS process with general fpras construction

• f Jerrum-Sinclair (1993):

◮ Let A ⊆ C0 ∪ C2 with π(A) ≥ 1/S(n) for some polynomial S(n) ◮ The following defines an fpras for π(A):

◮ Let R(G) be our upper bound for tmix(δ) with δ = ξ/[8S(n)] ◮ Let Y = 1A ◮ Run the PS process T = R(G) time steps and record YT ◮ Independently generate 72ξ−2S(n) such samples and take the

sample mean

◮ Repeat 6 lg⌈1/η⌉ + 1 such experiments and take the median

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V ◮ Pick uniformly random v ∼ u

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V ◮ Pick uniformly random v ∼ u

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

Introduction Main Theorem Proof Discussion

Prokofiev-Svistunov process

PS proposals:

◮ If A ∈ C0:

◮ Pick uniformly random u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

◮ If A ∈ C2:

◮ Pick random odd u ∈ V ◮ Pick uniformly random v ∼ u ◮ Propose A → A△uv

Metropolize proposals with respect to PS measure π(·)

Introduction Main Theorem Proof Discussion

Proof of rapid mixing

◮ We use the path method ◮ Consider the transition graph G = (V, E) of the PS process

◮ V = C0 ∪ C2 ◮ E = {AA′ : P(A, A′) > 0}

Introduction Main Theorem Proof Discussion

Proof of rapid mixing

◮ We use the path method ◮ Consider the transition graph G = (V, E) of the PS process

◮ V = C0 ∪ C2 ◮ E = {AA′ : P(A, A′) > 0}

◮ Specify paths γI,F in G between pairs of states I, F

Introduction Main Theorem Proof Discussion

Proof of rapid mixing

◮ We use the path method ◮ Consider the transition graph G = (V, E) of the PS process

◮ V = C0 ∪ C2 ◮ E = {AA′ : P(A, A′) > 0}

◮ Specify paths γI,F in G between pairs of states I, F ◮ If no transition is used too often, the process is rapidly mixing

Introduction Main Theorem Proof Discussion

Proof of rapid mixing

◮ We use the path method ◮ Consider the transition graph G = (V, E) of the PS process

◮ V = C0 ∪ C2 ◮ E = {AA′ : P(A, A′) > 0}

◮ Specify paths γI,F in G between pairs of states I, F ◮ If no transition is used too often, the process is rapidly mixing ◮ In the most general case, one specifies paths between each pair

I, F ∈ C0 ∪ C2

Introduction Main Theorem Proof Discussion

Proof of rapid mixing

◮ We use the path method ◮ Consider the transition graph G = (V, E) of the PS process

◮ V = C0 ∪ C2 ◮ E = {AA′ : P(A, A′) > 0}

◮ Specify paths γI,F in G between pairs of states I, F ◮ If no transition is used too often, the process is rapidly mixing ◮ In the most general case, one specifies paths between each pair

I, F ∈ C0 ∪ C2

◮ For PS process, convenient to only specify paths from C2 to C0

C2 C0

Introduction Main Theorem Proof Discussion

Proof of rapid mixing

◮ We use the path method ◮ Consider the transition graph G = (V, E) of the PS process

◮ V = C0 ∪ C2 ◮ E = {AA′ : P(A, A′) > 0}

◮ Specify paths γI,F in G between pairs of states I, F ◮ If no transition is used too often, the process is rapidly mixing ◮ In the most general case, one specifies paths between each pair

I, F ∈ C0 ∪ C2

◮ For PS process, convenient to only specify paths from C2 to C0

I F C2 C0

Introduction Main Theorem Proof Discussion

Lemma (Jerrum-Sinclair-Vigoda (2004))

Consider MC with state space Ω and stationary distribution π. Let S ⊂ Ω, and specify paths Γ = {γI,F : I ∈ S, F ∈ Sc}. Then tmix(δ) ≤ log   1 δ min

A∈Ω π(A)

 

• 2 + 4

π(S) π(Sc) + π(Sc) π(S)

• ϕ(Γ)

where ϕ(Γ) :=

• max

(I,F )∈S×Sc |γI,F |

• max

AA′∈E

      

• (I,F )∈S×Sc

γI,F ∋AA′

π(I)π(F) π(A)P(A, A′)       

Introduction Main Theorem Proof Discussion

Lemma (Jerrum-Sinclair-Vigoda (2004))

Consider MC with state space Ω and stationary distribution π. Let S ⊂ Ω, and specify paths Γ = {γI,F : I ∈ S, F ∈ Sc}. Then tmix(δ) ≤ log   1 δ min

A∈Ω π(A)

 

• 2 + 4

π(S) π(Sc) + π(Sc) π(S)

• ϕ(Γ)

where ϕ(Γ) :=

• max

(I,F )∈S×Sc |γI,F |

• max

AA′∈E

      

• (I,F )∈S×Sc

γI,F ∋AA′

π(I)π(F) π(A)P(A, A′)       

◮ We choose S = C2.

Introduction Main Theorem Proof Discussion

Lemma (Jerrum-Sinclair-Vigoda (2004))

Consider MC with state space Ω and stationary distribution π. Let S ⊂ Ω, and specify paths Γ = {γI,F : I ∈ S, F ∈ Sc}. Then tmix(δ) ≤ log   1 δ min

A∈Ω π(A)

 

• 2 + 4

π(S) π(Sc) + π(Sc) π(S)

• ϕ(Γ)

where ϕ(Γ) :=

• max

(I,F )∈S×Sc |γI,F |

• max

AA′∈E

      

• (I,F )∈S×Sc

γI,F ∋AA′

π(I)π(F) π(A)P(A, A′)       

◮ We choose S = C2. Elementary to show:

2 n mx mx + 1 ≤ π(C2) π(C0) ≤ n−1, and π(A) ≥ x 8 m for all A

Introduction Main Theorem Proof Discussion

Lemma (Jerrum-Sinclair-Vigoda (2004))

Consider MC with state space Ω and stationary distribution π. Let S ⊂ Ω, and specify paths Γ = {γI,F : I ∈ S, F ∈ Sc}. Then tmix(δ) ≤ log   1 δ min

A∈Ω π(A)

 

• 2 + 4

π(S) π(Sc) + π(Sc) π(S)

• ϕ(Γ)

where ϕ(Γ) :=

• max

(I,F )∈S×Sc |γI,F |

• max

AA′∈E

      

• (I,F )∈S×Sc

γI,F ∋AA′

π(I)π(F) π(A)P(A, A′)       

◮ We choose S = C2. Elementary to show:

2 n mx mx + 1 ≤ π(C2) π(C0) ≤ n−1, and π(A) ≥ x 8 m for all A

◮ Therefore:

tmix(δ) ≤

• log

8 x

• − log δ

m 3 + 1 mx

• 2 m n ϕ(Γ)

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

I F I△F

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

I F I△F

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F I△F

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F A0

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F A1

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F A2

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

A2

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

I

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

F

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

◮ Γ = {γI,F }

F

Introduction Main Theorem Proof Discussion

Choice of Canonical Paths

◮ To transition from I to F

◮ Flip each e ∈ I△F

◮ If (I, F) ∈ C2 × C0 then

I△F ∈ C2

◮ I△F = A0 ∪

• i≥1 Ai
• ◮ A0 is a path

◮ Ai disjoint cycles

I F

◮ γI,F defined by:

◮ Traverse A0 ◮ . . . then A1 ◮ . . . then A2 ◮ . . .

◮ Γ = {γI,F }

F

One can show that ϕ(Γ) ≤ ∆(G)n4m/2

Introduction Main Theorem Proof Discussion

Bounding the congestion

◮ Define measure Λ

Λ(A) = x|A|      n, A ∈ C0 2, A ∈ C2 1, A ∈ C4

◮ π(A) =

Λ(A) Λ(C0 ∪ C2) If T = AA′ is a maximally congested transition ϕ(Γ) =

• (I,F )∈P(T )

π(I)π(F) π(A)P(A, A′) max

(I,F )∈S×Sc |γI,F |

Introduction Main Theorem Proof Discussion

Bounding the congestion

◮ Define measure Λ

Λ(A) = x|A|      n, A ∈ C0 2, A ∈ C2 1, A ∈ C4

◮ π(A) =

Λ(A) Λ(C0 ∪ C2) If T = AA′ is a maximally congested transition ϕ(Γ) ≤

• (I,F )∈P(T )

π(I)π(F) π(A)P(A, A′)m

Introduction Main Theorem Proof Discussion

Bounding the congestion

◮ Define measure Λ

Λ(A) = x|A|      n, A ∈ C0 2, A ∈ C2 1, A ∈ C4

◮ π(A) =

Λ(A) Λ(C0 ∪ C2) If T = AA′ is a maximally congested transition ϕ(Γ) ≤

• (I,F )∈P(T )

Λ(I)Λ(F) Λ(A)P(A, A′) m Λ(C0 ∪ C2)

Introduction Main Theorem Proof Discussion

Bounding the congestion

◮ Define measure Λ

Λ(A) = x|A|      n, A ∈ C0 2, A ∈ C2 1, A ∈ C4

◮ π(A) =

Λ(A) Λ(C0 ∪ C2) If T = AA′ is a maximally congested transition ϕ(Γ) ≤ m Λ(C0 ∪ C2)

• (I,F )∈P(T )

Λ(I)Λ(F) Λ(A)P(A, A′)

Introduction Main Theorem Proof Discussion

Bounding the congestion

◮ Define measure Λ

Λ(A) = x|A|      n, A ∈ C0 2, A ∈ C2 1, A ∈ C4

◮ π(A) =

Λ(A) Λ(C0 ∪ C2)

◮ Define for each T = AA′ ∈ E

◮ ηT : C2 × C0 → C0 ∪ C2 ∪ C4 ◮ ηT (I, F) = I△F△(A ∪ A′)

If T = AA′ is a maximally congested transition ϕ(Γ) ≤ m Λ(C0 ∪ C2)

• (I,F )∈P(T )

Λ(I)Λ(F) Λ(A)P(A, A′)

Introduction Main Theorem Proof Discussion

Bounding the congestion

◮ Define measure Λ

Λ(A) = x|A|      n, A ∈ C0 2, A ∈ C2 1, A ∈ C4

◮ π(A) =

Λ(A) Λ(C0 ∪ C2)

◮ Define for each T = AA′ ∈ E

◮ ηT : C2 × C0 → C0 ∪ C2 ∪ C4 ◮ ηT (I, F) = I△F△(A ∪ A′)

◮

Λ(I)Λ(F) Λ(A)P(A, A′) ≤ 4∆(G)n Λ(ηT (I, F)) If T = AA′ is a maximally congested transition ϕ(Γ) ≤ m Λ(C0 ∪ C2)

• (I,F )∈P(T )

Λ(I)Λ(F) Λ(A)P(A, A′)

Introduction Main Theorem Proof Discussion

Bounding the congestion

◮ Define measure Λ

Λ(A) = x|A|      n, A ∈ C0 2, A ∈ C2 1, A ∈ C4

◮ π(A) =

Λ(A) Λ(C0 ∪ C2)

◮ Define for each T = AA′ ∈ E

◮ ηT : C2 × C0 → C0 ∪ C2 ∪ C4 ◮ ηT (I, F) = I△F△(A ∪ A′)

◮

Λ(I)Λ(F) Λ(A)P(A, A′) ≤ 4∆(G)n Λ(ηT (I, F)) If T = AA′ is a maximally congested transition ϕ(Γ) ≤ m Λ(C0 ∪ C2)

• (I,F )∈P(T )

Λ(I)Λ(F) Λ(A)P(A, A′) ≤ 4 ∆(G) n m 1 Λ(C0 ∪ C2)

• (I,F )∈P(T )

Λ(ηT (I, F))

Introduction Main Theorem Proof Discussion

Bounding the congestion

◮ Define measure Λ

Λ(A) = x|A|      n, A ∈ C0 2, A ∈ C2 1, A ∈ C4

◮ π(A) =

Λ(A) Λ(C0 ∪ C2)

◮ Define for each T = AA′ ∈ E

◮ ηT : C2 × C0 → C0 ∪ C2 ∪ C4 ◮ ηT (I, F) = I△F△(A ∪ A′)

◮

Λ(I)Λ(F) Λ(A)P(A, A′) ≤ 4∆(G)n Λ(ηT (I, F))

◮ ηT is an injection

If T = AA′ is a maximally congested transition ϕ(Γ) ≤ m Λ(C0 ∪ C2)

• (I,F )∈P(T )

Λ(I)Λ(F) Λ(A)P(A, A′) ≤ 4 ∆(G) n m 1 Λ(C0 ∪ C2)

• (I,F )∈P(T )

Λ(ηT (I, F))

Introduction Main Theorem Proof Discussion

Bounding the congestion

◮ Define measure Λ

Λ(A) = x|A|      n, A ∈ C0 2, A ∈ C2 1, A ∈ C4

◮ π(A) =

Λ(A) Λ(C0 ∪ C2)

◮ Define for each T = AA′ ∈ E

◮ ηT : C2 × C0 → C0 ∪ C2 ∪ C4 ◮ ηT (I, F) = I△F△(A ∪ A′)

◮

Λ(I)Λ(F) Λ(A)P(A, A′) ≤ 4∆(G)n Λ(ηT (I, F))

◮ ηT is an injection

If T = AA′ is a maximally congested transition ϕ(Γ) ≤ m Λ(C0 ∪ C2)

• (I,F )∈P(T )

Λ(I)Λ(F) Λ(A)P(A, A′) ≤ 4 ∆(G) n m 1 Λ(C0 ∪ C2)

• (I,F )∈P(T )

Λ(ηT (I, F)) ≤ 4 ∆(G) n mΛ(C0 ∪ C2 ∪ C4) Λ(C0 ∪ C2)

Introduction Main Theorem Proof Discussion

Bounding the congestion

◮ Define measure Λ

Λ(A) = x|A|      n, A ∈ C0 2, A ∈ C2 1, A ∈ C4

◮ π(A) =

Λ(A) Λ(C0 ∪ C2)

◮ Define for each T = AA′ ∈ E

◮ ηT : C2 × C0 → C0 ∪ C2 ∪ C4 ◮ ηT (I, F) = I△F△(A ∪ A′)

◮

Λ(I)Λ(F) Λ(A)P(A, A′) ≤ 4∆(G)n Λ(ηT (I, F))

◮ ηT is an injection

If T = AA′ is a maximally congested transition ϕ(Γ) ≤ m Λ(C0 ∪ C2)

• (I,F )∈P(T )

Λ(I)Λ(F) Λ(A)P(A, A′) ≤ 4 ∆(G) n m 1 Λ(C0 ∪ C2)

• (I,F )∈P(T )

Λ(ηT (I, F)) ≤ 4 ∆(G) n mΛ(C0 ∪ C2 ∪ C4) Λ(C0 ∪ C2) = 4 ∆(G) n m

• 1 +

Λ(C4) Λ(C0) + Λ(C2)

Introduction Main Theorem Proof Discussion

Bounding the congestion cont. . .

The final step is to note that the high-temperature expansion implies Λ(C4) Λ(C0) ≤ 1 n n 4

• so that

ϕ(Γ) ≤ 4 ∆(G) n m

• 1 + Λ(C4)

Λ(C0)

Introduction Main Theorem Proof Discussion

Bounding the congestion cont. . .

The final step is to note that the high-temperature expansion implies Λ(C4) Λ(C0) ≤ 1 n n 4

• so that

ϕ(Γ) ≤ 4 ∆(G) n m

• 1 + Λ(C4)

Λ(C0)

• ≤ 4 ∆(G) n m
• 1 + 1

n n 4

Introduction Main Theorem Proof Discussion

Bounding the congestion cont. . .

The final step is to note that the high-temperature expansion implies Λ(C4) Λ(C0) ≤ 1 n n 4

• so that

ϕ(Γ) ≤ 4 ∆(G) n m

• 1 + Λ(C4)

Λ(C0)

• ≤ 4 ∆(G) n m
• 1 + 1

n n 4

• ≤ 4 ∆(G) n mn3

8

Introduction Main Theorem Proof Discussion

Bounding the congestion cont. . .

The final step is to note that the high-temperature expansion implies Λ(C4) Λ(C0) ≤ 1 n n 4

• so that

ϕ(Γ) ≤ 4 ∆(G) n m

• 1 + Λ(C4)

Λ(C0)

• ≤ 4 ∆(G) n m
• 1 + 1

n n 4

• ≤ 4 ∆(G) n mn3

8 = ∆(G) n4 m 2

Introduction Main Theorem Proof Discussion

Bounding the congestion cont. . .

The final step is to note that the high-temperature expansion implies Λ(C4) Λ(C0) ≤ 1 n n 4

• so that

ϕ(Γ) ≤ 4 ∆(G) n m

• 1 + Λ(C4)

Λ(C0)

• ≤ 4 ∆(G) n m
• 1 + 1

n n 4

• ≤ 4 ∆(G) n mn3

8 = ∆(G) n4 m 2

Introduction Main Theorem Proof Discussion

Discussion

◮ Can we obtain sharper results if we focus on special families of

graphs, such as G = Zd

L?

Introduction Main Theorem Proof Discussion

Discussion

◮ Can we obtain sharper results if we focus on special families of

graphs, such as G = Zd

L? ◮ The PS process is closely related to a modification of the

“lamplighter walk” in which lamps are always switched when

• visited. Can we use this similarity to say something more precise

when G = Zd

L?

Introduction Main Theorem Proof Discussion

Discussion

◮ Can we obtain sharper results if we focus on special families of

graphs, such as G = Zd

L? ◮ The PS process is closely related to a modification of the

“lamplighter walk” in which lamps are always switched when

• visited. Can we use this similarity to say something more precise

when G = Zd

L? ◮ Study related spin models using similar methods?

Mixing time bound for PS process

Theorem (Collevecchio, G., Hyndman, Tokarev 2014+)

The mixing time of the PS process on graph G = (V, E) with parameter x ∈ (0, 1) satisfies tmix(δ) ≤

• log

8 x

• − log δ

m 3 + 1 m x

• ∆(G)m2n5,

with n = |V |, m = |E| and ∆(G) the maximum degree.

High-temperature expansions and the PS measure

◮ Let ∂A = {v ∈ V : v has odd degree in (V, A)}

High-temperature expansions and the PS measure

◮ Let ∂A = {v ∈ V : v has odd degree in (V, A)} ◮ Let CW := {A ⊆ E : ∂A = W} for W ⊆ V

High-temperature expansions and the PS measure

◮ Let ∂A = {v ∈ V : v has odd degree in (V, A)} ◮ Let CW := {A ⊆ E : ∂A = W} for W ⊆ V ◮ Let Ck :=

W ⊆V |W |=k CW for integer 1 ≤ k ≤ |V |

High-temperature expansions and the PS measure

◮ Let ∂A = {v ∈ V : v has odd degree in (V, A)} ◮ Let CW := {A ⊆ E : ∂A = W} for W ⊆ V ◮ Let Ck :=

W ⊆V |W |=k CW for integer 1 ≤ k ≤ |V |

◮ λ(·) defined by λ(S) = A∈S x|A| for S ⊆ {A ⊆ E}

High-temperature expansions and the PS measure

◮ Let ∂A = {v ∈ V : v has odd degree in (V, A)} ◮ Let CW := {A ⊆ E : ∂A = W} for W ⊆ V ◮ Let Ck :=

W ⊆V |W |=k CW for integer 1 ≤ k ≤ |V |

◮ λ(·) defined by λ(S) = A∈S x|A| for S ⊆ {A ⊆ E}

If x = tanh β then E(Ising)

G,β v∈W

σv

• = λ(CW )

λ(C0)

High-temperature expansions and the PS measure

◮ Let ∂A = {v ∈ V : v has odd degree in (V, A)} ◮ Let CW := {A ⊆ E : ∂A = W} for W ⊆ V ◮ Let Ck :=

W ⊆V |W |=k CW for integer 1 ≤ k ≤ |V |

◮ λ(·) defined by λ(S) = A∈S x|A| for S ⊆ {A ⊆ E}

If x = tanh β then E(Ising)

G,β v∈W

σv

• = λ(CW )

λ(C0)

◮ PS measure defined on the configuration space C0 ∪ C2

π(A) = λ(A) nλ(C0) + 2λ(C2)

• n,

A ∈ C0, 2, A ∈ C2.

High-temperature expansions and the PS measure

◮ Let ∂A = {v ∈ V : v has odd degree in (V, A)} ◮ Let CW := {A ⊆ E : ∂A = W} for W ⊆ V ◮ Let Ck :=

W ⊆V |W |=k CW for integer 1 ≤ k ≤ |V |

◮ λ(·) defined by λ(S) = A∈S x|A| for S ⊆ {A ⊆ E}

If x = tanh β then E(Ising)

G,β v∈W

σv

• = λ(CW )

λ(C0)

◮ PS measure defined on the configuration space C0 ∪ C2

π(A) = λ(A) nλ(C0) + 2λ(C2)

• n,

A ∈ C0, 2, A ∈ C2.

◮ Ising susceptibility χ =

1 π(C0)

◮ Ising two-point correlation function E(σuσv) = n

2 π(Cuv) π(C0)

High-temperature expansions and the PS measure

◮ Let ∂A = {v ∈ V : v has odd degree in (V, A)} ◮ Let CW := {A ⊆ E : ∂A = W} for W ⊆ V ◮ Let Ck :=

W ⊆V |W |=k CW for integer 1 ≤ k ≤ |V |

◮ λ(·) defined by λ(S) = A∈S x|A| for S ⊆ {A ⊆ E}

If x = tanh β then E(Ising)

G,β v∈W

σv

• = λ(CW )

λ(C0)

◮ PS measure defined on the configuration space C0 ∪ C2

π(A) = λ(A) nλ(C0) + 2λ(C2)

• n,

A ∈ C0, 2, A ∈ C2.

◮ Ising susceptibility χ =

1 π(C0)

◮ Ising two-point correlation function E(σuσv) = n

2 π(Cuv) π(C0)

◮ PS measure is stationary distribution of PS process